The "Doomsday Rule" is a mental algorithm used to calculate the day of the week for any given date. It was popularized by mathematician John Horton Conway, who created a simple method to determine the weekday without needing a calendar. The rule relies on memorizing specific anchor days for each century and using a series of calculations based on the date's month and year. To apply the Doomsday Rule, you first identify the anchor day for the century of the date. Then, you calculate the "doomsday" for the year, which is a specific day that falls on the same weekday for certain dates each year. By combining these elements, you can easily find the day of the week for any date.